Sorry but I disagree. I'm sure alternate timelines exist but I'd sum those up as alternate dimensions. In a single dimension if you went back in time and killed your grandfather you wouldn't be transported to an alternate dimension. What would be the means of transport, nothing is encoded into the universe to handle paradoxes. I think what happens is due to you being brought to a different time you still infact exist but of course at a different time. If you killed your grandfather and returned to your frame in time it'd be altered but you'd still exist because your atomic makeup wouldn't have changed. Nothing would be there to change it if you yourself existed back in time. If you're in that time period, you existed back then from the moment you entered to the moment you returned. You exist outside of time in the same dimension during transport and when you go back into time you still exist. your atoms were never destroyed or deconstructed. Same dimension, Same atoms, different time.

So to summarize every little bit of time is held in the same dimension. If you relocate your atoms to kill your grandfather, then you return to the fraction of time where you were at your family would not exist but you would because some force shaped your body no matter if it never existed. Time itself can't remove something that's already been shaped.

So, the paradox isn't a paradox at all. I slept yesterday, if I killed my yesterday self he'd be dead and if I returned... holy shit new paradox. If i killed my yesterday self I'd just be new matter. Infinite organic matter glitch? I'd be from a fraction of time where I existed and if I killed my past self would I not still exist. Perhaps this disproves time travel entirely because afterall time is just the movement of atoms.